Problem: Rewrite the function by completing the square. $f(x)= 2 x^{2} +13 x +20$ $f(x)=$
Explanation: $\begin{aligned} f(x)&=2 x^2 +13 x +20 \\\\ &=2 \left(x^2 +\dfrac{13}{2} x\right) +20 \end{aligned}$ Now we want to complete $x^2 +\dfrac{13}{2} x$ into a perfect square. To do that, we should add $\left(\dfrac{{\frac{13}{2}}}{2}\right)^2={\dfrac{169}{16}}$ to it: $x^2{+\dfrac{13}{2}}x+{\dfrac{169}{16}}=\left(x +\dfrac{13}{4}\right)^2$ We add ${\dfrac{169}{16}}$ inside the parentheses, and subtract ${2}\cdot{\dfrac{169}{16}}$ outside them, to keep the expression equivalent. $\begin{aligned} &\phantom{=}{2} \left(x^2 +\dfrac{13}{2} x\right) +20 \\\\ &={2}\left(x^2 +\dfrac{13}{2} x+{\dfrac{169}{16}}\right) +20 -{2}\cdot{\dfrac{169}{16}} \\\\ &=2 \left(x +\dfrac{13}{4}\right)^2 +20 -\dfrac{169}{8} \\\\ &=2 \left(x +\dfrac{13}{4}\right)^2 -\dfrac{9}{8} \end{aligned}$ In conclusion, the function after completing the square is written as: $f(x)=2 \left(x +\dfrac{13}{4}\right)^2 -\dfrac{9}{8}$